If G is a graph without isolated vertices, and if rand s are positive integers, then the (r, s)-domination number 'Yr,s(G) of G is the cardinality of a smallest vertex set D such that every vertex not in D is within distance r from some vertex in D, while every vertex in D is within distance s from another vertex in D. This generalizes the total domination number 'Yt(G) = 'Yl,l(G). Let T( G) de...

The bondage number b(G) of a graph G is the smallest number of edges whose removal results in a graph with domination number greater than the domination number of G. Kang and Yuan [Bondage number of planar graphs. Discrete Math. 222 (2000), 191198] proved b(G) 6 min{8,∆+ 2} for every connected planar graph G, where ∆ is the maximum degree of G. Later Carlson and Develin [On the bondage number o...

A set S of vertices of a graph G = (V ,E) without isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number γt (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) i...

In this paper, we introduce and do an initial investigation of a variant Grundy hop domination in graph called the connected domination. We show that number lies between graph. particular, give realization results involving domination, numbers. Moreover, determine numbers some graphs.

• Improved approximation algorithm for Budgeted Connected Dominating Set. Optimal Star Edge-Vertex Domination. We consider the version of classical Set problem (BCDS). Given a graph G and budget k , we seek connected subset at most vertices maximizing number dominated in . improve over previous ( 1 − e ) / 13 state art [Khuller, Purohit, Sarpatwar, SODA 2014 ] by introducing new method performi...

Let G = (V,E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the ...